Question

Find the remaining trigonometric functions of θ based on the given information.csc θ = 257 and cos θ < 0

Answer 1

Cos(θ) < 0, so we know it would be in Quadrant 2 or 3
then csc(θ) = 257, but csc(θ) = $\frac{1}{sin(θ)}$ = 257
==> sin(θ) = $\frac{1}{257}$
it is positive, so now we can determine that is in Quadrant 2

sin(θ) = opp./hyp both of opp and hyp are positive but adj suppose to negative because that way it leads the cos(θ) < 0
cos(θ) = adj/hyp
Pythereom to find the adj: $257^{2} - 1^{2} = adj ==\ \textgreater \ adj \sqrt{ 257^{2} - 1^{2} } ==\ \textgreater \ adj = 16 \sqrt{258}$

cosθ = $\frac{16 \sqrt{258} }{257}$
tanθ = $\frac{1}{16 \sqrt{258} }$
cosθ = $16 \sqrt{258}$